If you have back problems, the FIRST and FOREMOST important thing as far as buying a bike is that you need a good bike fit. Hardtail, full-suspension, whatever...those one-size-fits-all Walmart bikes will most likely not fit you, and all the playing with the seat post height and spacers under your bars won't make it fit as well as a bike with geometry actually suited to you. A well-fitting hard tail will be better to your back than an ill-fitting full-suspension.

Furthermore, the "suspension" on those bikes is crap, and it'll bounce you around like a pogo-stick. Most likely, your lower back muscles will tense up to compensate. The result? Back pain.

What you need isn't a junky full-suspension bike that doesn't fit you. Obviously it depends on the cause of your back pain, but there's a good chance that core work, targetted stretching, and a good bike fit will help.

ok, I forgot to mention... back pain is due to weight lifting, that I was too stupid to do in a gym back in 2009...

any type of suspension prevents recurring of my back pain... I know it from experience, I had seatpost suspension, it was good while it lasted... it was bouncy but it's exactly what I needed... shock went to suspension system in seatpost, and not to my spine...

and position... no worries... I'll install a higher handlebar if I need to, as I did for my previous hardtail...

the only thing that worries me is whether this frame will support my weight, that's all

The specs on that rocket are actually not too bad (its got deore lx, 7005 frame, and manitou suspension).
The other 2 seem lesser and for more $$$ at that. Throw on some smooth tires (like a specialized armadillo) and shell hold up perfectly fine for you city commute

Edit: I noticed the picture says small frame. If you're 6'0" yould want a large

The Rocket seems to have two different descriptions. I'd probably trust the one that says it has generic Shimano shifters and front derailer, and an Alivio rear derailer. I think the Delta that costs more is the best speced, though they all seem to have two different descriptions of what parts come on them... ? Even that one says it weighs 50lbs, though. Yeesh!

OP, if your suspension seat-post worked out fine, why not just get a new seat-post? Or if you need a new bike, get a hardtail and put a suspension post on it? It would be much more efficient riding for around town to have a hardtail with a squishy post than an entire squishy rear end, and especially with a cheap-o pogo stick "shock" in the rear. Also, it looks like those bikes come with big knobby tires. Not what you want for riding around town(unless you prefer riding harder, not smarter). If you planned on riding this on "real" MTB trails, I'd look at finding a better quality bike. At Least something with a decent rear derailer and shifters. Generally the generic Shimano stuff is OK for recreational riding on roads, but you'll be grinding gears and breaking/replacing stuff in no time if you try taking it on bumpy, rocky, rooty trails much. It does look like they come with a decent seat. I ran a WTB Speed V for a season on my MTB and it was totally decent. They're pretty popular, so I'd assume others agree. If you definitely want to get one of these Schwinns, at Least make sure it has double walled rims. ESPECIALLY at your weight, you'll trash single walls before you know what happened. Hell, I can't ride a single wall rim without trashing it and I weigh 150lbs.

The Rocket seems to have two different descriptions. I'd probably trust the one that says it has generic Shimano shifters and front derailer, and an Alivio rear derailer. I think the Delta that costs more is the best speced, though they all seem to have two different descriptions of what parts come on them... ? Even that one says it weighs 50lbs, though. Yeesh!

OP, if your suspension seat-post worked out fine, why not just get a new seat-post? Or if you need a new bike, get a hardtail and put a suspension post on it? It would be much more efficient riding for around town to have a hardtail with a squishy post than an entire squishy rear end, and especially with a cheap-o pogo stick "shock" in the rear. Also, it looks like those bikes come with big knobby tires. Not what you want for riding around town(unless you prefer riding harder, not smarter). If you planned on riding this on "real" MTB trails, I'd look at finding a better quality bike. At Least something with a decent rear derailer and shifters. Generally the generic Shimano stuff is OK for recreational riding on roads, but you'll be grinding gears and breaking/replacing stuff in no time if you try taking it on bumpy, rocky, rooty trails much. It does look like they come with a decent seat. I ran a WTB Speed V for a season on my MTB and it was totally decent. They're pretty popular, so I'd assume others agree. If you definitely want to get one of these Schwinns, at Least make sure it has double walled rims. ESPECIALLY at your weight, you'll trash single walls before you know what happened. Hell, I can't ride a single wall rim without trashing it and I weigh 150lbs.

it's double walled, yeah
I found out that weight capacity is 300 pounds for apparently all of full suspension schwinns
I'll get the one with 18 inch frame, because I need some extra space for mudguard and reflector
seat will be replaced, tires too... gears are ok for my recreational needs... if mechs break, I'll replace, I know how to and have the tools needed

the seatpost that I used broke, and I don't want to deal with those things anymore... it just give me headaches
whereas FS frames are designed for exactly that... absorbing shocks
and there's much to absorb here in my neck of woods with potholes left and right

why I don't want 1500 dollah specialised?
because it's new york... they will steal it in a heart beat...

I've been @235 most of my adult life. Frames hold up really well. Most will handle 300 pounds riders. Wheelsets should be more of a concern, weight can cause less expensive wheelsets to get out of round pretty quick.

Sizing a bike frame is very important. The hardtails you have tried may be small and could be causing muscle tension as well. You probably already know the importance of stretching and icing. Keep up stretching and icing, especially after a ride.

I am thinking that you need to get a frame that fits well. This is a good fit calculator.

Just adding a lame 2 cents - as to back pain (and yes I have a problem back too)... don't assume a full suspension bike will take out all of the bumps, hits, drops whatever f dirt trails. Now if you can spring for like a 8 inch travel bike or above, maybe, but otherwise a cross-country bike with full suspension still requires you to use your arms and legs as part of the suspension. I have a fairly nice X-C bike and there is little I ride where I can just sit and cruise over; I am most of the time back off the saddle.

Less expensive bikes really don't have any suspension to write home about so I don't think that's your answer. You would be better served by making sure the bike fits you properly and learning how to use your body to take the hits.

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